Magnetic Fields

Rotation Measure Synthesis

The radio emission of galactic and extragalactic radio sources often shows a significant amount of linear polarization. This linearly polarized signal can be utilized as an extremely effective probe of the intervening magneto-ionic medium between the source and observer since the intrinsic plane of polarization undergoes Faraday rotation as this emission propagates. The amount of that rotation is designated the Rotation Measure (RM) which is defined by:

$$\theta = \theta_{0} + RM \lambda^2$$ (1.16)

where $\theta$ is the observed and $\theta_0$ the intrinsic polarization angle, RM is in rad/m² and $\lambda$ is the wavelength in meters. The differential rotation $d\theta$ across a bandwidth, $d\nu$, centered at frequency $\nu$ is given by:

$$d\theta = 2~RM \lambda^2 d\nu / \nu$$ (1.17)

Sufficiently narrow frequency channels must be used to measure $\theta$, particularly at low frequencies, otherwise bandwidth depolarization will result. The RM depends on the properties of the medium as:

$$RM = 0.81 \int B_{\parallel}n_e(l) dl$$ (1.18)

where $B_{\parallel}$ is the longitudinal component of the magnetic field measured in $\mu$Gauss, n_e is the electron density in cm^-3 and l is measured in pc.

The $\lambda^2$ dependence of the polarization plane noted above has often been used to produce estimates of the magnetic field structure and electron density distribution in the vicinity of radio sources and within the intervening medium. Traditionally only a small number of observations of the polarization angle at discrete wavelengths have been obtained and a least squares fit has been used to calculate the RM. The utility of the RM probe can be vastly enhanced by using very large instantaneous bandwidths together with high spectral resolution. In this case a coherent addition of the polarized signals, $\vec{P}$=Q+jU, in the various bands can be formed as a function of the RM from:

$$\vec{I}(RM) = \int W(\lambda) \vec{P}(\lambda) e^{-2iRMd\lambda^2} d\lambda$$ (1.19)

where we have defined the frequency window function, W($\lambda$), which is determined by the locations, widths and shapes of the various frequency bands. This is effectively a Fourier Transform with respect to the RM and produces a cube with two spatial axes and a third of RM. The point spread function in RM is given by:

$$\vec{T}(RM) = \int W(\lambda) e^{-2iRMd\lambda^2} d\lambda$$ (1.20)

With a suitable choice of the function, W($\lambda$), a clean RM beam is achieved, providing both high RM resolution and a low sidelobe level. A practical example of this technique is given in Fig. 1.20 in which the RM in the direction of the pulsar J0214+4232 has been derived from only 8 frequency channels of 5 MHz width spread over the interval 319-380 MHz. Due to the very limited frequency sampling and uniform weighting in this example the RM sidelobe level is relatively high. Even so, the RM resolution determined by the inverse of the total observing bandwidth corresponds to about 12 rad/m². Given a substantial signal-to-noise ratio, the RM centroid can of course be determined to much higher precision than the RM beamwidth.

  

Figure 1.20: Plot of the polarized intensity as a function of RM for the pulsar J0214+4232.

The range of RM's which is accessible to a given observation depends on the observing wavelength, the channel bandwidth and the total observing bandwidth in addition to the instrumental sensitivity. Inserting the specifications for the SKA, we see that with a spectral coverage of 50% ( $\nu/\Delta\nu~=~2$) and a spectral resolution, $\nu/d\nu~=~10^4$, a one degree precision in the polarization angle would allow measurement of |RM| < 875 with an accuracy of about 0.02 rad/m² at a wavelength of 1 m and $\vert RM\vert~<~2.4\times10^5$ with 4.8 rad/m² precision at a wavelength of 6 cm.

An example of the application of this technique to a supernova remnant (SNR) is given by Gaensler et al (1998). With the sensitivity of the SKA accurate RMs will be obtainable for most SNRs. In conjunction with other information, this can be used as a (weak) indicator of distance.

This new tool should be particularly powerful in the study of weakly polarized extended sources, such as giant radio galaxies where the RMs are known to be small. Small changes in the RM across the surface of extended sources should be easily traceable, especially if these RMs show some spatial coherence.

Very small variations in RM can be expected to occur close to the core of AGN when polarized structure moves relative to a foreground Faraday screen. With a sensitivity of 0.02 rad/m², extremely sensitive measurements of the ionized gas in front of polarized radio sources can be made.

The observed RM is an integral along the line of sight. If the medium emitting the polarized waves is mixed with the medium rotating the wave vectors, the observed RM does not vary with $\lambda^2$ anymore (Sokoloff et al. 1998), and the observed RM cannot be used directly to compute magnetic field strengths. On the other hand, any variation of RM with wavelength contains valuable information about the rotating medium. A large number of frequency channels, as planned for the SKA, is required to check the wavelength dependence of RM.

Polarization Studies of the Interstellar Medium in the Galaxy and in Nearby External Galaxies

The bulk of the radio emission from the Galaxy at frequencies below 5 GHz is generated by the synchrotron process through the interaction of relativistic electrons with magnetic fields. Studies of the polarization state of that radiation gives us the opportunity to measure many properties of those fields, and the enhanced resolution and sensitivity that will become available with the SKA will open new opportunities. As synchrotron emission propagates through the intervening magneto-ionic medium its inherent linear polarization suffers Faraday rotation effects depending on both the field strength and the electron density, and measurements of Faraday rotation effects can give information on parameters which is unobtainable by other means. Lastly, Zeeman splitting of spectral lines from maser sources offers the possibility of direct field strength measurements under a variety of astrophysical conditions.

Our knowledge of the magnetic field configuration of the Galaxy is based on measurements of the Faraday rotation of signals from extragalactic sources as they propagate through the Galaxy (e.g. Simard-Normandin et al. 1981, Broten et al. 1988, Han et al. 1997). The deficiency of this method of probing the field has been the relatively small number of sources suitable for these measurements (at most a few hundred). The SKA will increase the number of sources available for this kind of work by a large factor, leading to a dramatic improvement in our detailed knowledge of the field configuration. It will also be possible to extend this field mapping technique to external galaxies. However, with the present telescopes only a handful of polarized background sources can be observed even in the most extended galaxy M31 (Han et al. 1998). With the SKA, the much larger number of suitable sources which could be seen through a face-on galaxy, such as M101, would allow a mapping of the field and, with the aid of other data, an estimate of the field strength that would complement information available from mapping of the synchrotron emission from the M101 itself. By this method even magnetic fields without detectable synchrotron emission together with thin ionized gas, e.g. in galactic halos, can be discovered. In a similar way Faraday rotation probing of SNRs (e.g. Kim et al. 1988) and other objects, such as clusters of galaxies (e.g. Kim et al. 1990), would be improved by a very large factor.

RM determinations using pulsar signals add to our knowledge of the Galactic field configuration, with the added advantage that we have some knowledge of pulsar distance, albeit with considerable errors. The SKA will contribute in this field too, largely through the discovery of many additional pulsars.

Numerous nearby galaxies have now been studied in some detail with angular resolutions from arcminutes to tens of arcseconds, yielding a resolution of some 100's of parsecs at the distance of galaxies in the Local Group. Accurate flux determinations at various frequencies can be used to deduce the magnetic field strength through the equipartition argument.

The mapping of linear polarization (at several frequencies to correct for Faraday rotation) has shown surprisingly well-aligned magnetic fields (Beck et al. 1996). In fact, magnetic fields commensurate with the spiral arm structure have been traced. However there are indications of deviations from the large-scale structures both in the spiral arms and in the centres of galaxies. In several galaxies regular fields are concentrated between the optical spiral arms. The sensitivity and high angular resolution of the SKA will play a key role in understanding these structures in view of the competing theories of the origin of the magnetic field.

There has been a resurgence of interest in the polarization of the Galactic synchrotron emission, usually referred to as the ``background'' emission, but actually in the foreground. After a lull since the 1970s due to a lack of improvement in instruments, the subject has been revived through the advent of sensitive systems capable of precise polarimetry over wide areas of the sky. Large regions of the sky have now been surveyed with resolution of the order of 10 arcminutes, in the Southern sky by Duncan et al. (1997) and in the Northern sky by Uyaniker et al. (1998 and 1999). Synthesis telescopes have also become capable of wide-field polarimetry, with good understanding of instrumental effects and calibration. This has enabled polarization mapping with arcminute resolution at 327 MHz with the Westerbork Telescope (Wieringa et al. 1993) and at 1420 MHz with the DRAO Synthesis Telescope (Gray et al. 1998 and 1999).

  

Figure 1.21: An image at $\lambda$21 cm of effects of the diffuse Faraday Screen in the Milky Way. This image from the Canadian Galactic Plane Survey shows the spatial variations in polarisation position angle imposed by changes in the structure of the Galactic magnetic field and diffuse electrons distribution on scales of 10's of pc. The SKA allow this technique to be used on other galaxies to investigate the structure and generation mechanisms of galactic magnetic fields.

Many effects are seen in the quoted papers which are predominantly effects in angle, since they have no significant counterpart in polarized intensity. Fig. 1.21 shows an example of such a region from the Canadian Galactic Plane Survey being made with the DRAO Synthesis Telescope. Such features are interpreted as the effects of a ``Faraday screen'', a region of ionized gas threaded by magnetic field lying between the source of the emission and the observer. Of course, Faraday rotation also occurs within the emitting volume, complicating interpretation.

Faraday rotation depends on the product of the electron density and the field strength. In a typical Galactic field of 3 $\mu$G, it is easy to detect the Faraday rotation produced by an ionized region in the foreground with an emission measure 1 cm^-6pc at 1.5 GHz. This is much higher sensitivity than most other methods of detecting ionized gas, which may be either an asset or an insuperable complication.

In certain circumstances sufficient information is available to allow the product of field and electron density to be separated. For example, Gray et al. (1999) have been able to measure the magnetic field strength in the extended ionised envelope of the HIIregion W4. The enhanced angular resolution of the SKA will permit such measurements to much greater distances, and with a vastly increased number of HIIregions.

The high density and the turbulent structure of the ionized gas in HII regions cause very high and very rapidly variable values of rotation measure through these objects, which effectively eliminates the polarization of synchrotron radiation propagating through them (by bandwidth or by beam depolarization). This effect can be exploited to limit the distance to polarization structures (Gray et al. 1998). This technique is analogous to the use of HIIregions to absorb the background synchrotron emission in order to measure the synchrotron emissivity in the foreground. With two orders of magnitude improvement in sensitivity and with the small beam of the SKA such studies at high frequencies (e.g. 5 GHz) can be extended to much greater distances (right across the Galaxy). Depolarization will be much less so that the polarized emission from distant regions can be studied.

Dickey (1997) has shown that HI absorption can be used to place limits on the distance to the polarized Galactic emission. Spectral-line mapping in Stokes Q and U removes all emission features because the HI emission is not polarized. Any absorption detected must arise from absorption of polarized continuum emission. With the sensitivity of the SKA, this technique can be extended to obtain absorption distances to other Galactic synchrotron emitters too faint for other methods. Examples include the non-thermal emission from WR-stars (Williams et al. 1997), from novae (Reynolds & Chevalier 1984), and from low-surface-brightness SNRs.

The SKA will play an important part in the study of SNRs. In addition to the detailed morphological information which will come from total-power images of excellent sensitivity and resolution, there is much to be learned from polarimetry of these objects. It is generally believed that cosmic-ray acceleration occurs at the shock fronts of SNRs, but many aspects of acceleration theory can only be improved through much more detailed knowledge of magnetic field structure at the shock front. Detailed examination of field structures in young SNRs will allow study of turbulent field amplification at the shock front. Detailed examination of the interaction of the shock fronts of older remnants with the clumpy interstellar medium (ISM) will show the role of the magnetic field in moderating these interactions, which are a significant source of energy input to the ISM.

The magnetic field must play an important role in the coupling of SN ejecta with the ISM, currently a poorly understood process. Such coupling can be seen in some SNRs (e.g. Cas A (Anderson & Rudnick 1995) and the Vela SNR (Aschenbach et al. 1995)) and the sensitivity and resolution of the SKA will permit such studies to be extended to other objects.

Finally, the very high sensitivity of the SKA will make it the premier instrument for the measurement of magnetic fields through the Zeeman effect. Measurements of the Zeeman effect require the detection of a very small difference between the signals received in the two senses of circular polarization and it is extremely difficult to account for all instrumental effects. Measurements using the HIline at 1420 MHz have been controversial, and it appears that measurements to date which have detected field strengths of the order of 10 $\mu$G are inconclusive. However, the method has proved useful in circumstances where the field is enhanced, for example in the shells of SNRs. Frail et al. (1994) and Claussen et al. (1997) have used Zeeman splitting of the 1720 MHz line of OH, excited into maser emission by the interaction of the SNR shock with molecular material, to measure fields of order 0.2 mG. The SKA has the potential to make Zeeman measurements at much lower field strengths.

An angular dimension of 0.1''; corresponds to only 0.3pc at the distance to M31, and magnetic fields are expected to be highly uniform on this scale in many regions of M31. The average field strength in M31 is only 5 MicroGauss, but on small scales we may expect 10 MicroGauss, enough to see the Zeeman effect. Galaxies with higher field strengths are e.g. NGC6946 and M51 with 10 MicroGauss on average, locally up to 20 MicroGauss. Here the spatial resolution of the SKA is a few pc, but uniform fields can be expected even on this allowing Zeeman measurements of the strength. The successful pursuit of these goals will require careful attention to the polarization performance of the instrument.